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Exceptional Polynomials and Monodromy Groups in Positive Characteristic

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Exceptional Polynomials and Monodromy Groups in Positive Characteristic Exceptional polynomials and monodromy groups in positive characteristic Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Julius-Maximilians-Universit¨at W¨urzburg vorgelegt von Florian M¨oller Institut f¨urMathematik der Universit¨atW¨urzburg W¨urzburg, M¨arz 2009 ii Danksagung Ich m¨ochte mich herzlich bei Professor Peter M¨ullerf¨ur seine Unterst¨utzung in allen Phasen meiner Promotion bedanken. Als Betreuer meiner Dissertation lastete auf Herrn M¨uller die Aufgabe, mir einerseits zu zeigen wie sch¨onfreie und eigenst¨andige Forschungst¨atigkeit sein kann, mir andererseits aber auch Probleme vorzulegen, die mich in meiner Promotion voranbrachten. Dies ist Herrn M¨uller ausgezeichnet gelungen. Besonders erw¨ahnen m¨ochte ich noch, dass Professor M¨uller mehrere komplizierte math- ematische Situationen, mit denen ich mich konfrontiert sah, stark vereinfachte. Erst durch seine Hilfe sind mir einige Beweise gelungen oder in ihre jetzige lesbare(re) Form gebracht worden. Weiterhin danke ich Professor Theo Grundh¨ofer,der f¨ur meine Fragen immer ein offenes Ohr hatte und mir an mancher Stelle mit wertvollen Literaturhinweisen half. Weiter habe ich ihm eine zus¨atzliche Anstellung am Lehrstuhl f¨urMathematik III zu verdanken. Ohne die Unterst¨utzung von Herrn Grundh¨ofer w¨are mir meine Promotion erheblich schwerer gefallen. Mein besonderer Dank gilt meiner Familie, die mich in meinem Vorhaben stets best¨arkte und mir einen Ausgleich zur mathematischen Besch¨aftigung bot. iii iv Contents 1. Introduction 1 I. General Results 5 2. Monodromy groups and affine polynomials 7 2.1. Polynomials and monodromy groups . 7 2.2. Affine groups and affine polynomials . 8 2.3. Generic polynomials . 10 3. Calculation of differents and the genus-0 condition 13 3.1. The general case . 13 3.2. Applications to the case of affine Galois groups . 18 3.2.1. Consequences of Hasse-Arf and Herbrand . 18 3.2.2. Bounds for the number of fixed points for elements of G . 20 3.2.3. Bounds for ind(∞)............................. 20 3.2.4. Bounds for ind(p) with p a finite place . 21 II. Application to specific problems 23 4. Affine polynomials with g ≤ 2 25 4.1. g =0......................................... 25 ∼ 4.1.1. Case (A): H = Cm is cyclic . 26 4.1.2. Case (B): H is an elementary abelian p-group . 28 ∼ 4.1.3. Case (C): H = (Cp × · · · × Cp) o Cm is a semidirect product . 28 4.1.4. Case (D): H is dihedral in even characteristic . 28 4.1.5. Case (E): H is dihedral in odd characteristic . 28 ∼ 4.1.6. Case (F): H = A5 in characteristic 3 . 28 4.1.7. Case (G): H =∼ PSL(2, q) with p 6=2 ................... 29 4.1.8. Case (H): H =∼ PGL(2, q) with p 6=2................... 37 4.1.9. Case (I): H =∼ PGL(2, 2m)......................... 40 4.2. g =1......................................... 41 4.2.1. Ramification in E|K(t) .......................... 41 4.2.2. Cases in odd characteristic p > 2..................... 43 4.2.3. Cases in even characteristic p =2..................... 43 4.3. g =2......................................... 47 4.3.1. Cases in odd characteristic p > 2..................... 47 4.3.2. Cases in even characteristic p =2..................... 51 5. AGL as a monodromy group 53 v Contents 6. Affine polynomials of degree p2 61 6.1. Cases in odd characteristic p 6=2......................... 62 6.2. Cases in even characteristic p =2......................... 63 7. Exceptional polynomials of degree p3 65 7.1. The cases in characteristic p > 3 ......................... 65 7.2. The cases in characteristic p ∈ {2, 3} ....................... 68 8. Exceptional polynomials of degree pr, r an odd prime, with 2-transitive group A 69 vi 1. Introduction In 1923 Schur [43] requested a description of all polynomials f ∈ Z[X] that induce a bijection on Z/pZ for infinitely many primes p. He proved that if f is of prime degree, then it is – up to q linear changes over the algebraic closure of Q – either a cyclic polynomial X or a Chebychev q 1 polynomial Tq(X) (defined implicitly by Tq(X + 1/X) = X + Xq ). He conjectured that in the general case f is a composition of such polynomials. This was proved by Fried [13] in 1970. Schur’s original question has been generalized in several ways: Turnwald [48] discusses the problem over integral domains, Guralnick, M¨uller, and Saxl [19] characterize rational functions r over number fields K so that r induces a bijection on the residue field Kp for infinitely many places p ∈ P(K). There is a different generalization of interest to us: we assume the base field to be of positive characteristic and search for exceptional polynomials, i.e. polynomials that fulfill the following Definition (Exceptionality) Assume k is a finite field. Let f ∈ k[X] be a polynomial with coefficients in k. If K is an extension field of k, then f is called a permutation polynomial over K if f induces a bijection on K. f ∈ k[X] is called exceptional over k if it is a permutation polynomial over infinitely many finite extensions of k. The classification of permutation polynomials has a long tradition and goes back to Her- mite [24] who noticed that any function k → k with k a finite field can be represented by a polynomial. A broad survey of permutation polynomials can be found in Lidl and Niederreiter [35]. An important step forward concerning the theory of exceptional polynomials was the refor- mulation of the original definition in terms of Galois theory and covering theory in positive characteristic. A first consequence was the proof of the following Exceptionality Lemma (cf. [14]) Let k be a finite field of characteristic p and t a tran- scendental over k. Let f ∈ k[X] be a polynomial. Denote the set of zeros of f(X) − t by Z. Fix a zero x ∈ Z. Then: (1) Suppose f is not a p-th power in k[X]. Then f is exceptional over k if and only if the x-stabilizer of the arithmetic monodromy group of f fixes no orbit of the x-stabilizer of the geometric monodromy group of f on Z \{x}. (A definition of monodromy groups is given on page 7.) (2) Suppose f is a composition f = f1 ◦ f2 with f1, f2 ∈ k[X]. Then f is exceptional over k if and only if both f1 and f2 are exceptional over k. In 1993 Fried, Guralnick, and Saxl [14] realized that this result reduces the classification of exceptional polynomials essentially to a question about primitive groups. They used the 1 1. Introduction Theorem of O’Nan-Scott and the classification of finite simple groups to obtain Theorem Let k be a finite field of characteristic p. Assume f ∈ k[X] is an indecomposable and exceptional polynomial of degree n. Denote the geometric monodromy group of f by G. Then one of the following holds: (1) n is an odd prime different from the characteristic p. The group G is cyclic or dihedral of degree n. r r r (2) n = p and G = Fp o H is an affine group with H ≤ GL(r, p) acting naturally on Fp. (3) p ∈ {2, 3}, there exists an odd integer a ≥ 3 with PSL(2, pa) ≤ G ≤ PΓL(2, pa), and 1 a a n = 2 p (p − 1). This classification solved in particular Carlitz’s conjecture (1966): if p is an odd prime, then the degree of f is odd. The three cases of the above theorem are understood with different degrees of completeness: Case (1) is classical; up to linear changes only cyclic or Chebychev polynomials of degree n arise here, cf. [38, Appendix]. This is the equivalent to Schur’s original conjecture. The first examples in case (3) were given by M¨uller [39] for p = 2 and a = 3. Cohen and Matthews [10] generalized these to an infinite series in even characteristic. Lenstra and Zieve [34] found a similar series for p = 3. Recently, Guralnick, Zieve, and Rosenberg [22, 20] gave a complete discussion of case (3). The polynomials occurring fulfill either G = PSL(2, pa) or G = PGL(2, pa). Case (2) is still open. The main problem is the difficulty to find restrictions for the group H; even the smallest possible case deg f = p requires some work, cf. [14, §5]. For some time the only examples of this case were additive polynomials and certain twists of them. In 1997 Guralnick and M¨uller [17] found a completely new series. Guralnick and Zieve [22] conjecture that there are no more polynomials belonging to this case. Up to the present every example fulfills the following Observation. The fixed field E of the affine kernel of the geometric monodromy group of f is rational. This observation motivates a classification of exceptional polynomials with primitive affine arithmetic monodromy group in terms of the genus g of E. This is done in chapter 4 up to g = 2. As a result, E is rational if and only if f belongs to a family of known polynomials. Moreover there are no exceptional polynomials with primitive affine arithmetic monodromy group for g ∈ {1, 2}. However, Theorem 4.25 shows that in case g = 2 the affine group AGL(2, 3) can be realized as the geometric monodromy group of a polynomial. Chapter 5 generalizes this AGL(2, 3)-polynomial; Theorem 5.7 eventually shows that every affine group AGL(r, pe) can be realized as the geometric monodromy group of a polynomial re of degree p . Unfortunately, these groups are 2-transitive on the r-dimensional Fpe -vector space; hence, this chapter does not offer exceptional polynomials at all.
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